Distributed knowledge: Difference between revisions

Revision as of 20:24, 5 October 2009

Revision as of 20:25, 5 October 2009

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The logicians Alice and Bob are sitting in their dark office wondering whether or not it is raining outside. Now, none of them actually knows, but Alice knows something about her friend Carol, namely that Carol wears her red coat [[only if]] it is raining. Bob does not know this, but he just saw Carol, and noticed that she was wearing her red coat. Even though none of them knows whether or not it is raining, it is ''distributed knowledge'' amongst them that it is raining. If either one of them tells the other what they know, it will be clear to the other that it is raining.

If we denote by <math>\varphi</math> that Carol wears a bluered coat and with <math>\varphi \implies \psi</math> that if Carol wears a red coat, it is raining, we have

: <math>(K_b\varphi \land K_a(\varphi \implies \psi)) \implies D_{a,b}\psi</math>